#384893

order	512
nilpotency class	3
isoclinism	isoclinic class of #31414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

863 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(5,6)(7,17)(8,22)(9,12)(10,16)(11,15)(13,20)(14,25)(18,26)(19,33)(21,29)(23,30)(24,27)(28,35)(31,34)(32,36);(6,12)(7,18,21,20)(8,23)(13,17,26,29)(15,16)(19,32,35,34)(24,25)(28,31,33,36);(5,7)(6,13)(8,19)(9,21)(10,18)(11,20)(12,26)(14,28)(15,29)(16,17)(22,31)(23,35)(24,32)(25,34)(27,33)(30,36);(5,8,11,25,9,23,10,24)(6,14,16,22,12,27,15,30)(7,19,20,34,21,35,18,32)(13,28,17,31,26,33,29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292089);